Busy-Time Scheduling on Heterogeneous Machines

We study a busy-time scheduling problem on heterogeneous machines (BSHM) which is motivated by server acquisition and task dispatching in cloud computing. The input of BSHM is a set of interval jobs, each specified by a size, an arrival time and a departure time. When a job arrives, it must be placed onto a machine immediately. The execution of a job cannot be interrupted until it departs. At any time, the total size of the jobs running on a machine cannot exceed the machine's capacity. m different types of machines are available and abundant machines are provided for each type. A type-i machine has a capacity g_i and is charged at a cost rate r_i when busy (running jobs). The target of BSHM is to schedule the given set of jobs onto machines with the minimum accumulated cost. Suppose the machine types are sorted by their capacities so that g₁ g₂ ≤ ⋯ ≤ g_m. We first consider two typical cases of BSHM. In BSHM-DEC, r_i/g_i ≥ r_i+1/g_i+1 holds for each i. In BSHM-INC, r_i/g_i ≤ r_i+1/g_i+1 holds for each i. For each case, we propose a O(1)-approximation algorithm in the offline setting and a O(μ)-competitive algorithm in the nonclairvoyant online setting. Finally, we discuss how the scheduling strategies developed for these two cases can be combined to deal with the general BSHM problem.